Chpter33 Percentge Contents: A Percentge B Expressing one quntity s percentge of nother C Finding percentge of quntity D The unitry method in percentge E Percentge increse nd decrese F Finding the originl mount G Simple interest H Compound interest (Extension)
44 PERCENTAGE (Chpter 3) Opening prolem A mnufcturer of rekfst cerels hs incresed the contents of their oxes y 20%. The new oxes now contin 1:5 kg of cerel. How much cerel did the old cerel oxes contin? 20% EXTRA! Whety Crunch 1.5 kg 20% EXTRA! Whety Crunch In this chpter we will explore some everydy pplictions of percentges, including percentge chnge, simple interest, nd compound interest. A PERCENTAGE The word percent mens out of every hundred. We therefore sy tht there is 100% in whole mount. A percentge is the comprison of portion with the whole mount. We use the symol % to show percentge. For exmple, 10% mens 10 out of every 100 or 10 25, nd 25% mens 100 100. So, x% = x 100. CONVERTING FRACTIONS AND DECIMALS TO PERCENTAGES To convert frction or deciml to percentge, we multiply y 100%. Since 100% = 1, this does not chnge the vlue of the numer. Exmple 1 Convert to percentge: 5 8 5 8 = 5 8 100% =62:5% 0:065 =0:065 100% =6:5% 0:065 c 1:35 c 1:35 =1:35 100% = 135% Shifting the deciml point 2 plces to the right multiplies y 100. EXERCISE 3A.1 1 Write these frctions s percentges: 1 2 13 c 2 5 20 g 31 50 h 21 40 i 3 8 d 17 25 5 j 16 23 e 100 k 101 125 f 14 10 l 2 1 8
PERCENTAGE (Chpter 3) 45 2 Write these decimls s percentges: 0:13 0:35 c 0:03 d 0:9 e 0:06 f 2:7 g 0:2 h 3:25 i 1:05 j 3:41 k 0:075 l 1:225 3 Write s percentge: 0:3 1 c 0:0004 d 3 2 e 5 6 f 3:027 g 25 16 h 17 i 5 4 5 j 0:7 CONVERTING PERCENTAGES TO FRACTIONS AND DECIMALS To convert percentge to frction, we write the percentge s frction with denomintor 100, then express the frction in simplest form. Exmple 2 Express s frction in simplest form: 150% 0:5% 150% = 150 100 150 50 = 100 50 = 3 2 0:5% = 0:5 100 = 0:5 2 100 2 = 1 200 =1 1 2 To convert percentge to deciml, we shift the deciml point 2 plces to the left. Exmple 3 Express s deciml: 43% 32 1 2 % 43% = 43. 100 =0:43 32 1 2 % = 32:5 100 =0:325 Shifting the deciml point 2 plces to the left divides y 100. EXERCISE 3A.2 1 Copy nd complete: 7% = 100 87% = 100 c 45% = 100 d 33% = 100
46 PERCENTAGE (Chpter 3) 2 Express s frction in simplest form: 20% 150% c 75% d 50% e 310% f 8% g 100% h 2% i 12 1 2 % j 1:5% k 87 1 2 % l 331 3 % 3 Express s deciml: 44% 39% c 95% d 60% e 125% f 400% g 50% h 225% i 0:7% j 1:02% k 6:75% l 50 1 4 % B EXPRESSING ONE QUANTITY AS A PERCENTAGE OF ANOTHER To express one quntity s percentge of nother, we first mke sure they re written with the sme units. We then write the first quntity s frction of the second, nd multiply y 100%. Exmple 4 Express s percentge: Simon drnk 200 ml of 1:25 L ottle of soft drink. 200 ml out of 1:25 L = 200 ml out of 1250 ml fwrite with the sme unitsg = 200 1250 100% = 16% So, Simon drnk 16% of the soft drink. EXERCISE 3B 1 Express s percentge: 15 cm out of 20 cm 11 minutes out of 25 minutes c 38 mrks out of 40 mrks d 18 kg out of 30 kg. 2 Write s percentge: 220 ml out of 2 L 24 minutes out of 1 hour c 35 cm out of 1:4 m d 18 months out of 2 1 2 yers. 3 On the tennis court, Mri hit 180 of 300 serves in. Write this s percentge. 4 Lily scored 76 mrks out of 80 for her Physics test. Wht ws her percentge? 5 Decln ws lte for home group 18 dys out of 50 lst term. Wht percentge of the dys ws he on time? 6 During the soccer seson, Mrk conceded 25 gols out of 150. Wht percentge of gols did he: concede sve?
PERCENTAGE (Chpter 3) 47 7 In chess tournment, Len won 14 mtches nd lost 2. Wht percentge of mtches did he: win lose? 8 Pete hd $5 in his pocket nd spent 65 cents on postge stmp. Wht percentge of his money did he spend? 9 Leil drnk 450 ml of 1:5 L ottle of wter. Wht percentge of the wter did she drink? 10 On his wy to work, Ben trvels 12:5 km y trin, nd the remining 750 m on foot. Wht percentge of the totl distnce is trvelled on foot? C FINDING A PERCENTAGE OF A QUANTITY To find percentge of quntity, we convert the percentge to deciml, nd then multiply. FINDING A PERCENTAGE OF A QUANTITY Exmple 5 Find: 1:5% of $2200 108% of 5000 kg (in tonnes) The word of indictes tht we should multiply the numers. 1:5% of $2200 =0:015 $2200 = $33 108% of 5000 kg =1:08 5 tonnes = 5:4 tonnes EXERCISE 3C 1 Find: 55% of $8 20% of $70 c 15% of 12 L d 250% of 40 kg e 3% of 3 tonnes f 37 1 % of 900 m g 8:5% of $3500 h 2:4% of 65 g 2 i 3 1 2 % of 375 ml j 71:5% of 18 m/s k 36% of 2 1 hours (in minutes) 2 2 Jillin scored 70% in her test out of 60 mrks. How mny mrks did she score? 3 In his lst snooker mtch, Neil successfully potted 96% of his long shots. If he ttempted 75 long shots, how mny did he get in? 4 In sell seson, Derek hit the ll 25:5% of the time. If he cme up to t 196 times, how mny hits did he hve? 5 A petroleum compny clims tht crs using their new premium unleded fuel will trvel 112% of the distnce trvelled on regulr unleded. If Geoff cn trvel 584 km on tnk of regulr fuel, how fr should he e le to trvel using the premium unleded?
48 PERCENTAGE (Chpter 3) 6 Nicki is prt of syndicte tht owns rcehorses. He owns 28% of Peter Pn, who lst week won $18 600 in the dery. How much should Nicki receive from this win? 7 A resturnt hs 10% surchrge on the ill on pulic holidys. If Jck nd Jill uy food nd wine to the vlue of $54 on pulic holidy, how much will they hve to py: for the surchrge in totl? D THE UNITARY METHOD IN PERCENTAGE Michelle nd Brigette own usiness. Brigette receives 25% of the profits ech month. Lst month, Brigette received $2080. How cn she work out the totl profit mde y the usiness lst month? The unitry method cn e used to solve this prolem. We first find 1%, then multiply y 100 to find the whole mount. Exmple 6 Find 100% of sum of money if 25% is $2080. 25% of the mount is $2080 ) 1% of the mount is $2080 25 = $83:20 ) 100% of the mount is $83:20 100 ) the whole mount is $8320 The unitry method cn lso e used to find other percentges of quntity. Exmple 7 18% of the students t school re in Yer 8, nd 16% of the students re in Yer 9. If the school hs 126 Yer 8 students, determine the numer of Yer 9 students. 18% is 126 students 126 ) 1% is 18 =7students ) 16% is 7 16 = 112 students So, the school hs 112 Yer 9 students. EXERCISE 3D 1 Find 100% if: 10% is 40 ml 45% is 225 g c 6% is $72 d 70% is 49 kg e 53% is $159 f 95% is 38 minutes 2 22% of students t school ride their ikes to school. If 132 students ride their ikes to school, how mny students ttend the school?
3 Find 40% of n injection if 5% of the injection is 7 ml. Find 84% of pcket of nuts if 14% of the pcket is 21 g. c Find 72% of the contents of ottle if 9% of the contents is 80 ml. d Find 18% of wge if 54% of the wge is $630. PERCENTAGE (Chpter 3) 49 4 An lloy contins 15% mngnese nd 85% iron. If 37:5 kg of mngnese is used to mke the lloy, how much: lloy is produced iron is used? 5 Approximtely 75 000 people in the Northern Territory re of indigenous origin, mking up 32:5% of the Northern Territory popultion. Estimte, to the nerest thousnd, the popultion of the Northern Territory. 6 When Vivin kes cookies, she lwys urns 20% of them. How mny cookies does Vivin need to ke so tht she finishes with 28 unurnt cookies? 7 When churning crem to mke utter, 48% of the crem comes out s utter, nd the reminder s uttermilk. How much utter will you get from 3:5 kg of crem? Amy needs 120 g of utter for recipe. How much crem would she need to churn? 8 Mddie opertes n ice crem vn. The tle longside shows the sles for ech flvour in the lst week, expressed s percentge of the totl numer sold. Wht percentge of ice crems sold were strwerry flvoured? Given tht Mddie sold 124 vnill ice crems, determine how mny: i chocolte ice crems were sold ii mint ice crems were sold. c Determine the totl numer of ice crems sold. Flvour % of sles Chocolte 36% Vnill 31% Strwerry 22% Mint 7:5% Bnn 3:5% Totl 100% 9 The children t cmp ech hd to choose one of cnoeing, hiking, or swimming for their fternoon ctivity. 45% of the children chose cnoeing, 18 children chose hiking, nd 15 children chose swimming. How mny children were t the cmp? 10 Mny food nd drink pckges come with lels like the one elow. ENERGY 870 kj DI* 10% PROTEIN 4.5 g DI* 9% FAT 0.7 g DI* 1% SAT FAT 0.25 g DI* 1% CARBS 27.9 g DI* 9% SUGARS 9.5 g DI* 11% SODIUM 115 mg DI* 5% For exmple, this food contins 4:5 g of protein, which is 9% of the recommended Dily Intke (DI) of protein. Use this informtion to find the recommended Dily Intke of: energy protein c sturted ft d sodium.
50 PERCENTAGE (Chpter 3) E PERCENTAGE INCREASE AND DECREASE Chnges in quntities re often expressed s percentge of the originl quntity. For exmple: ² Clss sizes hve incresed y 10%. ² House prices hve dropped y 12%. ² Cron emissions need to fll y 25% y the yer 2020. There re two different methods which cn e used to solve percentge increse or decrese prolems. METHOD 1: Step 1: Step 2: WITH TWO STEPS Find the size of the increse or decrese. Apply the chnge to the originl quntity y ddition or sutrction. Exmple 8 A fruit grower picked 1720 kg of pples lst yer. This yer she expects her crop to e 20% igger. How mny kilogrms of pples does she expect to pick this yer? Step 1: size of increse = 20% of 1720 kg =0:2 1720 kg = 344 kg Step 2: new mount = 1720 + 344 kg = 2064 kg The fruit grower expects to pick 2064 kg. EXERCISE 3E.1 1 Perform these opertions using two steps: increse $30 y 10% decrese 50 kg y 20% c increse 60 my25% d decrese 400 Ly1% e increse 50 000 people y 2:3% f decrese 45 minutes y 4%. 2 Ktie grows spinch in her grden. Lst yer she hrvested 6 kg. This yer her hrvest incresed y 30%. How much extr spinch did she hrvest this yer? Find the totl weight of Ktie s hrvest this yer. 3 Lst yer there were 2075 South Austrlin students enrolled in Yer 12 Physics. Enrolments decresed y 4% this yer. Find the decrese in enrolments. How mny students re enrolled this yer?
METHOD 2: WITH ONE STEP USING A MULTIPLIER PERCENTAGE (Chpter 3) 51 To increse n mount y 15%, we strt with 100% of the mount, then dd 15% of the mount to it. We now hve 100% + 15% = 115% of the mount. So, to increse n mount y 15% in one step, we multiply the mount y 115% or 1:15. The vlue 1:15 is clled the multiplier. To decrese n mount y 15%, we strt with 100% of the mount, then sutrct 15% of the mount from it. We now hve 100% 15% = 85% of the mount. So, to decrese n mount y 15% in one step, we multiply the mount y 85% or 0:85. The vlue 0:85 is clled the multiplier. Exmple 9 Increse 312 kg y 22%, giving your nswer to 3 significnt figures. Decrese $183 y 7%. To increse y 22%, we multiply y 100% + 22% = 122%, which is 1:22. ) the new mount =1:22 312 kg ¼ 381 kg To decrese y 7%, we multiply y 100% 7% = 93%, which is 0:93. ) the new mount = 0:93 $183 = $170:19 EXERCISE 3E.2 1 Find the multiplier corresponding to: n increse of 25% decrese of 10% c decrese of 19% d n increse of 8:2% e decrese of 7 1 % f n increse of 150%. 2 2 Use multiplier to perform the following, giving nswers to 3 significnt figures where pproprite: increse $50:40 y 20% decrese 46 cm y 13% c decrese 230 kg y 55% d increse 35 minutes y 7:5% e decrese 81 L y 4:5% f increse $67 y 250%. 3 In 1996, the totl popultion of Tsmnin devils in the wild ws estimted t 130 000. However, the trnsmissile Devil Fcil Tumour Disese hs cused 70% decrese in numers. Estimte the popultion of Tsmnin devils in the wild tody. 4 Ed is 160 cm tll, nd his sister Peggy is 8:5% tller thn he is. How tll is Peggy? 5 In the yer 2000, it ws estimted tht there were 361 million Internet users worldwide. In the next ten yers, tht numer incresed y 445%. Estimte the numer of Internet users worldwide in 2010.
52 PERCENTAGE (Chpter 3) 6 Jessi s rent increses y 5% t the end of ech yer. If she currently pys $220 per week, how much will Jessi e pying two yers from now? 7 On Mondy, the vlue of Rijken Industries shres were $7:50. The price incresed y 6% on Tuesdy, decresed y 1:5% on Wednesdy, nd decresed y 2% on Thursdy. Find the vlue of ech shre when trde opened on Fridy. FINDING A PERCENTAGE CHANGE If we know tht the mount of quntity chnges, the multiplier for the chnge is clculted y multiplier = new mount originl mount. We cn then use the multiplier to determine the percentge chnge. Exmple 10 Determine the percentge chnge when: 50 kg is incresed to 70 kg $160 is decresed to $120. multiplier = new mount originl mount 70 kg = 50 kg =1:4 This corresponds to 40% increse. multiplier = new mount originl mount = $120 $160 =0:75 This corresponds to 25% decrese. EXERCISE 3E.3 1 Find the percentge chnge when: $20 is incresed to $22 80 ml is decresed to 68 ml c 45 g is decresed to 27 g d 90 cm is incresed to 1:35 m. A multiplier greter thn 1 shows n increse. A multiplier less thn 1 shows decrese. 2 Descrie the percentge chnge in the following situtions: The price of hircut lst month ws $30. It hs since risen to $34:50. 1500 people ttended the Christms crols lst yer. It rined this yer, so only 1080 ttended. c Arthur ought house two yers go for $320 000. It is now worth $360 000. d e At her school s sports dy, Csey threw the jvelin 56:33 m, eting the previous school record of 52:40 m. John completed hlf-mrthon in 1 hour nd 52 minutes, improving on his previous est time of 2 hours nd 8 minutes.
3 Hrriot relly loves snkes. For her irthdy in 2009, she ws given pet crpet python y her uncle. She mesured its length on her irthdy ech yer: PERCENTAGE (Chpter 3) 53 AUS09-3-53 Yer 2009 2010 2011 2012 Length 58 cm 79 cm 94 cm 1:07 m 55X34 Clculte the percentge y which the snke s length incresed from: i 2009 to 2010 ii 2010 to 2011 iii 2011 to 2012. For the 3-yer period, clculte the overll percentge increse. 4 The following tle shows the numer of rod crsh deths in Austrlin sttes nd territories in My 2010 nd My 2011: NSW QLD SA TAS VIC WA ACT NT My 2010 38 22 12 5 31 22 3 3 My 2011 26 25 11 2 30 9 1 2 c Clculte the percentge chnge in rod deths from My 2010 to My 2011 for ech stte nd territory. Which stte or territory hd the: i lrgest percentge increse ii lrgest percentge decrese in rod deths? Find the percentge chnge in rod deths for the whole of Austrli. F FINDING THE ORIGINAL AMOUNT Given n originl mount nd certin percentge chnge, we hve seen how to otin new mount. It is often useful to e le to solve the reverse prolem. If we know the percentge chnge nd the new mount, we wnt to clculte the originl mount. We multiplied the originl mount y the multiplier to otin the new mount. Reversing this process, we divide the new mount y the multiplier to otin the originl mount. Exmple 11 An electricl goods store uys TV set t wholesle price. The price of the TV is incresed y 25% for sle y the store. Its selling price is now $550. For wht price did the store uy the TV set? cost price multiplier = selling price ) cost price 1:25 = $550 f100% + 25% = 125% = 1:25g ) cost price = $550 1:25 = $440 So, the television set cost the store $440.
54 PERCENTAGE (Chpter 3) EXERCISE 3F 1 Find the originl mount given tht: fter n increse of 20%, the length ws 24 cm fter decrese of 15%, the mss ws 51 kg c fter n increse of 3:6%, the mount ws $129:50 d fter n increse of 130%, the cpcity ws 9200 L e fter decrese of 0:8%, the ttendnce ws 49 600 people. We divide the new mount y the multiplier to find the originl mount. 2 In 2011, the Adelide Fringe Festivl hd 747 events. The festivl director sid this ws n increse of 66% since 2006. How mny events were t the 2006 Adelide Fringe Festivl? 3 A clothing store is hving 15% off sle. The price of cot hs een reduced to $119. How much does the cot usully cost? 4 Answer the Opening Prolem on pge 44. 5 From July to Septemer, the numer of people reding the Melourne newspper The Age dropped y 5:3% to 650 000. Estimte, to the nerest thousnd, the numer reding The Age in July. 6 Jon hs just received n electricity ill of $569:75. This is 32:5% more thn her previous ill. How much ws Jon s previous ill? Discussion ² If n mount is incresed y 18%, nd the resulting mount is then decresed y 18%, do we return to the originl mount? ² A new model cr is 18% more expensive thn the model it replces. Wht percentge discount must slesmn give in order to sell the new model cr t the old model price? G SIMPLE INTEREST When person orrows money from lending institution such s nk or finnce compny, the orrower must repy the lon in full, nd lso py n dditionl interest pyment. This is chrge for using the institution s money. Similrly, when money is invested in nk, the nk pys interest on ny deposits. The mount of interest pid is usully given s percentge of the mount which is orrowed or invested.
PERCENTAGE (Chpter 3) 55 SIMPLE INTEREST Simple interest is interest tht is clculted ech yer s fixed percentge of the originl mount of money orrowed or invested. The fixed percentge is clled the interest rte, nd is usully given per nnum which mens per yer. Suppose $8000 is invested for 5 yers t 10% per nnum simple interest. The simple interest pid for 1 yer = 10% of $8000 =0:1 $8000 = $800 The simple interest pid for 2 yers = 10% of $8000 2 =0:1 $8000 2 = $1600. The simple interest pid for 5 yers = 10% of $8000 5 =0:1 $8000 5 = $4000 These oservtions led to the simple interest formul. SIMPLE INTEREST FORMULA The simple interest I cn e clculted using the formul: I = Pin where P is the principl, or initil mount orrowed or invested, i is the flt rte of interest per nnum (p..), n is the time or durtion of the lon in yers. Exmple 12 Find the simple interest pyle on n investment of $20 000 t 12% p.. over period of 4 yers. Write the interest rte s deciml. P = 20 000 i = 12% = 12 100 = 0:12 n =4 Now I = Pin ) I = 20 000 0:12 4 ) I = 9600 ) the simple interest is $9600. In some res of finnce, sums of money my e invested over period of months or dys. However, the interest rte is still normlly quoted per nnum, so the time period n in the formul must e converted to yers.
56 PERCENTAGE (Chpter 3) Exmple 13 Clculte the simple interest pyle on n investment of $15 000 t 8% p.. over 9 months. P = 15 000 i =8%=8 100 = 0:08 n = 9 12 =0:75 Now I = Pin ) I = 15 000 0:08 0:75 ) I = 900 ) the simple interest is $900. EXERCISE 3G 1 Find the simple interest pyle on n investment of: $3000 t 11% p.. for 2 yers $5500 t 7 1 % p.. for 5 yers 2 c $15 000 t 5:75% p.. for 4 1 2 yers d $25 000 t 6:2% p.. for 10 yers. 2 Find the simple interest pyle on lon of: $2000 t 1:4% p.. over 9 months $8500 t 8% p.. over 3 months c $2 750 000 t 7% p.. over 18 months d $60 000 t 3:25% p.. over 10 months. Rememer to convert the time period to yers. Exmple 14 Determine the simple interest pyle on n investment of $100 000 t 15% p.. from April 28th to July 4th. From April 28th there re 2 dys left in April. ) 2 dys left in April we exclude the first dy (April 28) 31 dys in My 30 dys in June 4 dys in July we include the lst dy (July 4) 67 dys We divide the numer of dys y 365 to give the period in yers. P = 100 000 i = 15% = 15 100=0:15 n = 67 365 Now I = Pin ) I = 100 000 0:15 67 365 ) I = 2753:42 fto the nerest centg So, the interest pyle is $2753:42. 3 Gemm deposits $7800 in specil investment ccount on My 6th. The ccount pys 8:75% p.. simple interest. If Gemm withdrws the money on Septemer 9th, how much will her investment hve erned over this time?
PERCENTAGE (Chpter 3) 57 4 Wrren Brr deposited $7500 on July 1st in specil investment ccount which erns 11% p.. simple interest. On Septemer 19th he deposited nother $4900 in the ccount. He closed the ccount on Decemer 20th y withdrwing the totl lnce. In totl, how much interest hs his investment erned? Find the totl mount Wrren hs withdrwn. Exmple 15 Todd orrows $6000 for 5 yers t 11% p.. simple interest. Clculte the totl mount he must repy. If Todd repys the lon in equl monthly repyments over the 5 yers, find the mount he must repy ech month. P = 6000 i = 11% = 11 100 = 0:11 n =5 Now The totl mount to e repid = initil mount + interest = $6000 + $3300 = $9300 Todd repys the lon over 5 yers =60months. ) ech monthly repyment = $9300 60 = $155 I = Pin = 6000 0:11 5 = 3300 5 $4000 is orrowed under simple interest terms. Find the totl mount to e repid fter: 10 yers t 7% p.. 6 yers t 9 1 % p.. c 8 months t 13% p.. 2 6 Alice orrows $4700 from finnce compny to uy her first cr. The rte of simple interest is 17% nd she orrows the money over 5 yer period. Find: the totl mount Alice must repy the finnce compny her equl monthly repyments. 7 Cmeron orrows $15 000 from nk to renovte his house. He orrows the money t 9% p.. simple interest over 8 yers. Wht re his monthly repyments? 8 An electric guitr with ll ttchments is dvertised t $2400. Ben cn fford to py deposit of $600, ut he then hs to orrow the reminder t 12% p.. simple interest over 3 yers. Wht re his monthly repyments?
58 PERCENTAGE (Chpter 3) H COMPOUND INTEREST (EXTENSION) A more common method for clculting interest is compound interest. If you leve your money in the nk for period of time, the interest is utomticlly dded to your ccount. Compound interest llows you to ern interest on interest! With compound interest, ny interest tht is dded to your ccount will lso ern interest in the next time period. Compound interest is clculted s percentge of the totl mount t the end of the previous compounding period. In this course we will only consider interest which compounds ech yer. Suppose $1000 is plced in n ccount erning interest t rte of 10% p.. The interest is llowed to compound itself for three yers. We sy it is erning 10% p.. compound interest. We cn show this in tle: Yer Amount t eginning of yer Compound interest Amount t end of yer 1 $1000 10% of $1000 = $100 $1000 + $100 = $1100 2 $1100 10% of $1100 = $110 $1100 + $110 = $1210 3 $1210 10% of $1210 = $121 $1210 + $121 = $1331 After 3 yers there is totl of $1331 in the ccount. We hve erned $331 in compound interest. If we construct similr tle for $1000 in n ccount erning 10% p.. simple interest for 3 yers, we cn compre the two different types of interest. Yer Amount t eginning of yer Simple interest Amount t end of yer 1 $1000 10% of $1000 = $100 $1000 + $100 = $1100 2 $1100 10% of $1000 = $100 $1100 + $100 = $1200 3 $1200 10% of $1000 = $100 $1200 + $100 = $1300 After 3 yers there is totl of $1300 in the ccount, so we hve erned $300 in simple interest. Compring the two: Yer Compound interest Simple interest 1 $100 $100 2 $110 $100 3 $121 $100 Simple interest remins constnt. Compound interest increses over time.
PERCENTAGE (Chpter 3) 59 Exmple 16 How much interest will I ern if I invest $10 000 for 3 yers t: 15% p.. simple interest 15% p.. compound interest? We use the simple interest formul with P = 10 000, i =0:15, n =3. Now I = Pin ) I = 10 000 0:15 3 = 4500 Thus, the simple interest is $4500. Yer Initil mount Interest Finl mount 1 $10 000 15% of $10 000 = $1500:00 $11 500:00 2 $11 500 15% of $11 500 = $1725:00 $13 225:00 3 $13 225 15% of $13 225 = $1983:75 $15 208:75 The compound interest = finl mount initil mount = $15 208:75 $10 000 = $5208:75 EXERCISE 3H.1 1 $25 000 is invested t 6% p.. compound interest. Use tle to find: the finl mount fter 3 yers how much interest ws erned in the 3 yer period. 2 Clculte the interest erned on n investment of $2000 for 3 yers t: 5% p.. simple interest 5% p.. compound interest. 3 Use tle to determine the interest erned for the following investments: c $4000 t 7% p.. compound interest for 2 yers $500 t 2:5% p.. compound interest for 3 yers $11 000 t 5% p.. compound interest for 4 yers. Investigtion Compound interest spredsheet A spredsheet llows us to construct compound interest tle very rpidly. Wht to do: 1 Suppose $5000 is invested t 4% p.. compound interest for 10 yers. Open new spredsheet nd enter the following: SPREADSHEET
60 PERCENTAGE (Chpter 3) Highlight the formule in row 6. Fill down to row 14 for the 10th yer of the investment. How much interest is pid in: i yer 1 ii yer 10? How much is in the ccount fter: i 5 yers ii 10 yers? 2 Suppose $15 000 is invested t 6% p.. compound interest for 10 yers. Enter 15 000 in B1 nd 0:06 in B2. How much interest is pid in: i yer 1 ii yer 10? How much is in the ccount fter: i 5 yers ii 10 yers? 3 How long would it tke for $8000 invested t 5% p.. compound interest to doule in vlue? Hint: Enter 8000 in B1, 0:05 in B2, nd fill down further. 4 Wht compound interest rte is needed for $12 000 to doule in vlue fter 6 yers? Hint: Enter 12 000 in B1 nd repetedly chnge the interest rte in B2. THE COMPOUND INTEREST FORMULA Suppose you invest $1000 in the nk for 3 yers, erning 10% p.. compound interest. Your investment increses in vlue y 10% ech yer, so its vlue t the end of the yer is 100% + 10% = 110% of the vlue t the strt of the yer. This corresponds to multiplier of 1:1. After one yer your investment is worth $1000 1:1 = $1100 After two yers your investment is worth $1000 1:1 1:1 = $1000 (1:1) 2 = $1210 After three yers your investment is worth $1000 (1:1) 2 1:1 = $1000 (1:1) 3 = $1331 This suggests tht if the money ws left in your ccount for n yers, it would mount to $1000 (1:1) n. This leds us to the compound interest formul: F v = P v (1 + i) n where F v is the future vlue P v is the present vlue or originl mount i is the nnul interest rte s deciml (1 + i) is the multiplier n is the numer of yers of the investment. Notice tht the formul for F v ove gives the totl future vlue, which is the originl mount plus interest. To find the interest only we use: Compound interest = F v P v
PERCENTAGE (Chpter 3) 61 Exmple 17 Wht will $5000 invested t 8% p.. compound interest mount to fter 2 yers? How much interest is erned? An interest rte of 8% indictes tht i =0:08. For 2 yers, n =2 nd so F v = P v (1 + i) n = $5000 (1:08) 2 = $5832 Interest erned = $5832 $5000 = $832 EXERCISE 3H.2 1 $6000 is invested for 4 yers t 8% p.. Wht will this investment mount to if the interest is clculted s: simple interest compound interest? 2 $7500 is orrowed for 10 yers t 11% p.. How much will hve to e repid if the interest is clculted s: simple interest compound interest? 3 Wht will n investment of $3000 t 9% p.. compound interest mount to fter 5 yers? Wht prt of this is interest? 4 How much compound interest is erned y investing $20 000 for 6 yers t 4:5% p..? 5 You hve $8000 to invest for 3 yers. You hve een offered two investment options: Option 1: Invest t 9% p.. simple interest. Option 2: Invest t 8% p.. compound interest. Clculte the mount ccumulted t the end of the 3 yers for oth options nd decide which option to tke. Would you chnge your decision if you were investing for 5 yers? Review set 3 1 Express s deciml: 57% 375% 2 Write s percentge: 0:002 9 40 3 Express s percentge: 2 1 hours out of 10 hours 88 cm out of 12 metres. 2 4 Jillin hs repid $700 of $1500 det. Wht percentge of the det hs een repid? 5 Terry hs memorised 60% of piece of music. If the piece of music tkes 5 minutes to ply, how long is the section tht Terry hs memorised?
62 PERCENTAGE (Chpter 3) 6 Jess needs 30% of continer of sugr for her recipe. The continer is presently 75% full, nd this is 60 g of sugr. Wht weight of sugr does Jess need for her recipe? 7 A hospitl hs 36 doctors, who mke up 24% of the totl hospitl stff. How mny people work t the hospitl? 8 On Tuesdy, petrol ws eing sold t 132:5 cents per litre. On Wednesdy, the price ws 121:9 cents per litre. Descrie the percentge chnge in the price. 9 As result of cyclone, the price of nns rose 85% to $9:99 per kg. How much did nns cost efore the cyclone? 10 A footll stdium hs mximum cpcity of 60 000 people. One stnd is currently under renovtion however, so the stdium s cpcity hs een reduced y 8:5%. How mny people cn the stdium currently hold? 11 As prt of rod sfety cmpign, fines for ll trffic offences will increse y 5%. Copy nd complete the tle longside, showing the chnges to ech fine. Offence Old fine New fine Speeding $200 Drink driving $840 Not wering setelt $273 Illegl prking $52:50 12 Find the simple interest pyle on lon of $4000 t 7% p.. for 5 yers. 13 Rj orrows $5000 from nk t 8 1 % p.. simple interest. He orrows the money over 2 6 yer period. Find the totl mount tht Rj must repy the nk. Find Rj s equl monthly repyments. 14 Wht will n investment of $8000 t 6% p.. compound interest mount to fter 4 yers? Wht prt of this is interest? Prctice test 3A Click on the icon to otin this printle test. Multiple Choice PRINTABLE TEST Prctice test 3B Short response 1 Convert 62 1 % to: deciml frction in simplest form. 2 2 Convert 1 5 to percentge. 16 3 In n ice hockey mtch, Roerto sved 17 of 20 shots on gol. Wht percentge of shots did he sve? 4 Kte hd $20 in her wllet. While witing t the irport, she spent $4:50 on lrge coffee. Wht percentge of her money did she spend?
PERCENTAGE (Chpter 3) 63 5 Mrco hs 300 ml of red wine simmering in pn. He reduces the volume to 60% to mke suce. How much suce will he get? 6 Nikki scored 82% in her lst test. Her sister Bree scored 76%. If Bree scored 38 mrks: how mny mrks did Nikki score how mny mrks were in the test? 7 An nt is crrying 5000% of its own ody weight. If the lef it is crrying weighs 1 g, how much does the nt weigh? 8 The infltion rte is currently 3:3%. If chocolte r costs $1:60 now, how much will it cost next yer? 9 A soufflé increses in height y 125% while in the oven. If the finished height is 13:5 cm, how tll ws the soufflé efore it ws put in the oven? 10 Drren orrows $15 000 from the nk to extend his house. The nk will chrge him simple interest of 6:8% p.. for 5 yer lon. Find the totl mount Drren will need to repy. Prctice test 3C Extended response 1 The following tle shows the numer of irths in 2008 nd in 2009 in Austrli: NSW QLD SA TAS VIC WA ACT NT 2008 93 238 62 072 19 937 6683 69 976 31 418 4718 3903 2009 89 911 64 829 19 452 6531 70 820 30 377 5605 3715 Find, correct to one deciml plce, the percentge chnge in irths for ech stte nd territory. Which stte or territory hd the: i lrgest percentge increse ii lrgest percentge decrese? c Find the percentge chnge in irths for the whole of Austrli. 2 A mnufcturer sells refrigertors to n electricl store. The prts nd lour cost the mnufcturer $500. If the mnufcturer wishes to dd 40% profit, how much does the electricl store py for the refrigertor? The electricl store sells the refrigertor for $910 excluding GST. i How much profit does the store mke? ii If 10% GST must e dded to the price of the refrigertor, how much does consumer py? 3 Al orrows $10 000 from the nk to uy solr pnels for his house. He orrows the money t 7% p.. simple interest for 5 yers. Clculte the interest pyle on Al s lon. In totl, how much money will Al py ck? c Al wishes to py ck the lon in equl monthly instlments. i How mny repyments will there e? ii Wht is the size of ech repyment?
64 PERCENTAGE (Chpter 3) 4 Hlfwy through the 2011 AFL seson, the ldder looked s follows: Position Tem P W L D F A % 1 Geelong Cts 12 12 0 0 1226 858 142:89 2 Collingwood 11 10 1 0 1312 736 178:26 3 Crlton 12 9 2 1 1212 874 138:67 4 Hwthorn 12 9 3 0 1215 914 132:93 5 West Cost Egles 12 8 4 0 1182 959 123:25 6 Sydney Swns 12 7 4 1 991 950 104:32 7 Fremntle 12 6 6 0 1070 1171 91:37 8 Essendon 12 5 6 1 1214 1055 115:07 9 Melourne 12 5 6 1 1109 1085 102:21 10 Richmond 12 5 6 1 1192 93:49 11 North Melourne 12 5 7 0 1141 1123 101:60 12 St Kild 12 4 7 1 948 1035 13 Western Bulldogs 12 4 8 0 1014 1155 87:79 14 Adelide 12 3 9 0 935 1197 78:11 15 Brisne Lions 12 2 10 0 946 1223 77:35 16 Port Adelide 12 2 10 0 1358 69:22 17 Gold Cost Suns 11 2 9 0 792 1471 53:84 c Wht percentge of mtches hd Fremntle won? Wht percentge of mtches hd Hwthorn lost? The % column shows the points scored for tem, expressed s percentge of the points scored ginst the tem. Find: i St Kild s percentge ii the totl numer of points scored for Port Adelide iii the totl numer of points scored ginst Richmond. 5 A fst food resturnt sells soft drinks. The percentge of ech type sold in one week is shown in the tle. Wht percentge of drinks sold were lemonde? If the store sold 4000 drinks, how mny were: i col ii rsperry? c If 697 rsperry drinks were sold severl weeks lter, estimte the numer of diet col drinks sold. d Wht hve you ssumed in finding your nswer to c? Drink % of sles Col 35:9% Diet Col 15:2% Ornge 12:6% Lemonde Rsperry 8:5% Totl 100%